Variation Functions

Direct, inverse and joint variation all turn on one fixed number, the constant of variation. Recover it from a single matching pair of values and every other pair follows.

--

Some quantities rise together, some fall as the other rises, and some depend on two things at once. Variation gives each of these patterns an equation built around a single fixed number — the constant of variation — which you can recover from one pair of matching values and then reuse for every other pair.

Concept The three kinds of variation
x y
Direct — grows as grows
Inverse — shrinks as grows
Joint — varies with and together

In every case the number is the constant of variation, and .

Type Equation Proportion Graph
Direct Straight line through the origin
Inverse Hyperbola
Joint A flat surface
Theorem Why the proportion works

Because is the same for every pair of values, two pairs from the same relationship can be set equal to each other — and never has to be calculated.

Direct variation gives , so
Inverse variation gives , so

Notice the inverse proportion multiplies where the direct one divides. That single difference is what turns a rising line into a falling curve.

Example Direct variation

varies directly with , and when . Find when .

Set up the proportion:
Cross multiply:
Divide by :
when

Here , so the relationship is . Substituting directly gives the same .

Example Joint variation

varies jointly with and , and when and . Find when and .

Set up the proportion:
Simplify the products:
Cross multiply:
Divide by :
when and

Joint variation is direct variation with two partners instead of one — the product simply takes the place of .

Example Inverse variation

varies inversely with , and when . Find when .

Set up the proportion:
Simplify:
Divide by :
when

The value of grew five times larger, so became five times smaller. That trade-off is the signature of inverse variation, and here stays fixed throughout.

Summary
  1. Direct variation is — a straight line through the origin.
  2. Inverse variation is — a hyperbola, falling as rises.
  3. Joint variation is , direct variation with two partners.
  4. The constant is fixed for the whole relationship and can never be zero.
  5. Direct problems divide across the proportion; inverse problems multiply.
  6. One matching pair of values is enough to answer any other pair.
283. Relations and Functions
← Back to Semester